Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2009, Article ID 387317, 14 pages
doi:10.1155/2009/387317
Research Article
Trajectory Sensitivity Method and
Master-Slave Synchronization to Estimate
Parameters of Nonlinear Systems
Elmer P. T. Cari, Edson A. R. Theodoro, Ana P. Mijolaro,
Newton G. Bretas, and Luis F. C. Alberto
Departamento de Engenharia Elétrica da Escola de Engenharia de São Carlos,
Universidade de São Paulo 13566-590, São Carlos, SP, Brazil
Correspondence should be addressed to Luis F. C. Alberto, lfcalberto@usp.br
Received 1 February 2009; Revised 21 May 2009; Accepted 12 June 2009
Recommended by Elbert E. Neher Macau
A combination of trajectory sensitivity method and master-slave synchronization was proposed to
parameter estimation of nonlinear systems. It was shown that master-slave coupling increases the
robustness of the trajectory sensitivity algorithm with respect to the initial guess of parameters.
Since synchronization is not a guarantee that the estimation process converges to the correct
parameters, a conditional test that guarantees that the new combined methodology estimates the
true values of parameters was proposed. This conditional test was successfully applied to Lorenz’s
and Chua’s systems, and the proposed parameter estimation algorithm has shown to be very
robust with respect to parameter initial guesses and measurement noise for these examples.
Copyright q 2009 Elmer P. T. Cari et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Several algorithms make use of synchronization to estimate parameters of nonlinear systems
based on measured data. In 1, 2, for example, a Lyapunov-based design control and
synchronization were used to estimate parameters of nonlinear systems. A similar approach
that combines synchronization and geometric control is shown in 3. Other attempts,
including synchronization as an auxiliary tool for parameter estimation, are found in 4–6.
In addition, parameter estimation of delay systems can be found in 7, 8.
The measured output of the real system and the calculated output of an auxiliary
system, usually taken as the mathematical model of the real system, are compared. Based on
the output mismatch, parameters of the model are updated. When the outputs synchronize,
that is, the difference between the real system and the model is sufficiently small, the
parameters of the auxiliary system are assumed to be close enough to the real measurements.
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However, synchronization is not sufficient to ensure the correct parameter estimation.
In 9, for example, the authors follow the same techniques used in 6 and verify using
examples that even when the outputs synchronize, the model parameters are far from the
real system parameters.
Some studies try to guarantee the correct parameter estimation when the real system
and the model outputs synchronize. In 10, for example, the authors prove, for a class of
dynamical systems and using a convenient Lyapunov Control Function, that the system
globally synchronizes, and synchronization implies convergence to the true parameters if
some special conditions are satisfied.
In this paper, master-slave synchronization framework combined with a trajectory
sensitivity-based fitting algorithm is used to estimate parameters of nonlinear systems.
Trajectory sensitivity method has important characteristics and advantages when compared
to other approaches, such as Lyapunov Control Function framework. Trajectory sensitivitybased parameter estimation approach has the advantage of being easily implemented for
any nonlinear system while other approaches like Lyapunov-based metholodoly require the
existence of a Lyapunov control function. Moreover, Lyapunov-based design may require
long time intervals of measured data in order to achieve synchronization while trajectory
sensitivity approach can easily deal with short time intervals of measured data 1. Another
interesting feature of trajectory sensitivity approaches is that they provide estimates of the
initial conditions a desired estimate in some problems like the determination of the initial
state of population density of a living species in evolutionary studies that cannot be obtained
by the Lyapunov-based approach 1, 11.
In spite of these advantages, it is very difficult to guarantee synchronization and
convergence to the true parameters when trajectory sensitivity analysis is used as a fitting
algorithm. Usually, these assumptions are not checked, and numerical tests are used to
verify the robustness of the algorithm. In general, the estimation fails due to the high
relative sensitivities of the trajectories with respect to parameters and initial conditions that
leads to a very small convergence region of the algorithm. In this scenario, master-slave
synchronization emerges as a good auxiliary tool to increase the robustness of the parameter
estimation algorithm with respect to the initial guess of parameters.
Moreover, for this combined approach, there will be proposed a conditional test to
ensure convergence to the true parameter values. That is, if the parameter fitting algorithm
provides synchronization in a different sense, as it will explained in Section 3.1, then the
convergence to the true parameters is guaranteed. Based on the trajectory sensitivity method
and master-slave synchronization, some applications to estimate parameters of synchronous
generator in electric power system have been developed by the authors 12, 13.
The structure of this paper is as follow: Section 2 presents a general framework to
estimate parameters based on measured data and synchronization. In Section 3, there will be
exhibited the trajectory sensitivity method. Following that, master-slave synchronization and
trajectory sensitivity method will be combined and tested in two nonlinear systems; Sections
3.1 and 3.2. Moreover, the assumptions previously proposed to guarantee convergence to the
true parameters will be checked for these examples. The results will be discussed in Section 4.
2. General Framework to Estimate Parameters Based on
Measured Data and Synchronization
A common problem that appears in many applications is the necessity of estimating
parameters of a system based on the information contained in measured data time series.
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3
Figure 1 illustrates the traditional framework to estimate parameters of a general nonlinear
dynamical system based on measured data. In order to accomplish the estimation, an
auxiliary system, which is usually taken as the mathematical model of the real system, is
employed. Some of the measurements are chosen as inputs of the auxiliary system while the
others are used as outputs for comparison. Using the stored measured data of the inputs,
the outputs of the auxiliary system are calculated for an initial guess of parameters. Both the
measured and the auxiliary system outputs are compared. Based on the output mismatch,
the parameters are updated according to some prescribed rule.
The success of the general framework of Figure 1 to estimate parameters is guaranteed
if the following assumptions hold.
A1 Synchronization between the real system and the auxiliary system outputs should
imply parameters of the auxiliary system sufficiently close to parameters of the real
system.
A2 The parameter fitting algorithm provides output synchronization.
Assumption A1 is a necessary but not enough property, that has to be satisfied to
guarantee the correctness of estimates. Assumption A2 has to be satisfied to guarantee the
convergence of the algorithm.
There are many alternatives to design parameter fitting algorithms; however, in most
of the practical cases, it is very difficult to prove that assumption A2 holds. In general, the
success of the parameter estimation algorithm depends on personal experience to choose
a convenient auxiliary system and an efficient parameter fitting algorithm. Usually, the
effectiveness of the algorithms is numerically checked by means of a large number of tests,
and their problems are usually related to two main aspects.
1 The convergence region of the parameter estimation algorithm, that is a subset of
the synchronization region, can be very small; that is, if a good guess of parameters
is not available, then the parameter estimation algorithm diverges synchronization
is not achieved or converges to wrong values assumption A1 is not satisfied.
2 They may not be robust with respect to the presence of noise in the measurements
synchronization is not achieved due to noise presence.
In this paper, an unilateral coupling between the real and the auxiliary systems is
used to enhance the robustness of synchronization. As a result, parameters of nonlinear
systems including chaotic systems are correctly estimated, even for bad initial guesses of
parameters, in the presence of measurements with noise. The coupling is of unilateral type
14, that is, some of the outputs of the real system are used as inputs to the auxiliary system.
This type of coupling is known as master-slave synchronization. In this case, the real system
is the master while the auxiliary system is the slave. Figure 2 illustrates the situation.
It is important to emphasize that the authors that use control Lyapunov function
approach to adjust parameters have shown conditions to guarantee the satisfaction of
condition A2 for some class of systems 14. In our case, the adjustment is accomplished by
the sensitivity trajectory method using Newton’s method; therefore it is not possible to exhibit
an analytical condition to guarantee the satisfaction of condition A2. As a consequence,
we pursued another goal; that is, we offered a conditional proof of condition A2; that is,
A1 is numerically checked, and if synchronization is achieved, then A2 is true. Although
the control Lyapunov approach can provide, for some classes of nonlinear systems, the
satisfaction of assumptions A1 and A2, the motivation to study theses conditions under
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Mathematical Problems in Engineering
Input
data
Output data
w w1 , w2 Real system
−
Auxiliary system
y y1 , y2 Parameter fitting
algorithm
Figure 1: Framework to estimate parameters of nonlinear dynamical systems. Measured outputs of the
real system are compared to the outputs of an auxiliary system. Based on the output mismatch, the fitting
algorithm updates the parameters. If outputs are synchronized, parameters of the auxiliary system are
assumed to be sufficiently close to parameters of the real system model.
Input
data
Real system
Output data
w w1 , w2 −
Auxiliary system
y y1 , y2 Parameter fitting
algorithm
Figure 2: Parameter estimation framework with master-slave coupling between the real and the auxiliary
systems. Some outputs of the real system are used as inputs of the auxiliary system.
the sensitivity trajectory method is that this method can be implemented for any nonlinear
system in the form of 3.1-3.2, even for those we cannot exhibit a Lyapunov function.
In the next sections, this master-slave coupling approach is tested with a trajectory
sensitivity-based parameter fitting algorithm.
3. Trajectory Sensitivity Method with Master-Slave Coupling
In this section, trajectory sensitivity analysis and master-slave synchronization will be used
to estimate parameters of two chaotic systems. Although assumption A2 cannot be easily
checked, trajectory sensitivity analysis provides, from the practical point of view, advantages
that justify its usage.
Trajectory sensitivity method can easily deal with hard nonlinearities. A very
interesting extension of sensitivity method for differential algebraical equations DAE
systems subject to nonsmooth events, like switchings, is presented in 15.
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5
Consider a nonlinear system modeled by
d
xt f xt, p, ut ,
dt
yt g xt, p, ut ,
3.1
3.2
where xRn is the state vector, yRm is the output vector, uRl is the input vector, and pRk is
the parameter vector. Functions f and g are nonlinear, continuous, and Lipschitz with respect
to x, p, and u. Let pi be the ith component of p. We assume that f and g are differentiable
with respect to every component pi of p. In case they are not differentiable, a numerical
approximation can be used to evaluate these derivatives 16. The trajectory sensitivities
∂xt/∂pi and ∂yt/∂pi are computed, respectively, as
d ∂xt ∂f xt, p, ut
dt ∂pi
∂x
∂yt ∂g xt, p, ut
∂pi
∂x
∂xt ∂f xt, p, ut
·
,
∂pi
∂pi
∂xt ∂g xt, p, ut
·
.
∂pi
∂pi
3.3
Trajectory sensitivities quantify the variation of a trajectory with respect to small
variations in parameters. This quantification is used to update the model parameters in order
to minimize the distance between the outputs of the real system and the mathematical model.
The parameter fitting algorithm is formulated as an optimization problem; that is, we
try to minimize the error function J, which is given by
1
J p 2
To
y−w
t y − w dt,
3.4
0
where, w is the output vector of the real system, y is the output of the auxiliary system 3.2,
and 0, To is the time interval considered in the analysis. In other words, the minimization
searches for output synchronization.
Given an initial value p po , this optimization problem can be solved computing the
sensitivity ∂Jp/∂p and using a least squares method:
T
o
∂J p
∂y t G p y − w dt
∂p
0 ∂p
.
3.5
ppi
Expanding Gp in Taylor series around p pi and neglecting high-order terms one has
G p ≈ G pi ΓΔp,
3.6
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Mathematical Problems in Engineering
where Γp ∂Gp/∂p. Then Δp −Γ−1 Gpi . Parameters are fitted for the ith iteration
by pi1 pi Δpi1 . Matrix Γp can be computed by differentiating 3.5 and neglecting
higher-order terms, thus
Γ p ≈
∂y t ∂y dt
0 ∂p ∂p
T
3.7
ppi
When the measurements are sampled at discrete time intervals, the previous integrals are
replaced by summations. For more details see reference 17.
The error function indirectly depends on the parameters. Its evaluation requires
the solution of a set of ordinary differential equations the model. Usually, nonlinear
dynamic models do not have closed analytical solutions; that is, the evaluation of the error
function is made via numerical integration algorithms. Newton’s method, which is used
to achieve synchronization, demands the evaluation of trajectory sensitivity equations that
are also obtained via numerical integration algorithms. As far as we know, there is no
general condition to guarantee convergence of Newton’s algorithm or similar one in this
case.
The use of trajectory sensitivities in the framework described in Figure 1 has two
main problems: i the trajectory sensitivity method is very sensitive to initial conditions
first parameter guess, and ii parameters with very low sensitivities as compared to
other parameters are not numerically identifiable due to ill-conditioned calculations. These
problems become worst especially when many parameters have to be simultaneously
estimated and/or chaotic behavior is present. Very often, the trajectory sensitivity method
leads to erroneous estimations or even divergence of the numerical algorithm.
In this section, a master-slave coupling and trajectory sensitivity-based approach are
combined to estimate parameters of chaotic systems. Comparisons between the traditional
trajectory sensitivity approach presented in Figure 1 and the proposed methodology, which
includes a master-slave coupling to enhance synchronization robustness, are made in order
to show the advantages of the proposed methodology.
3.1. Lorenz’s System
Consider Lorenz’s system as the real system,
ẋ1 −σr x1 σr x2 ,
ẋ2 −x2 − x1 x3 rr x1 ,
3.8
ẋ3 −br x3 x1 x2 ,
and suppose that at least two states can be measured. Let w x1 , x2 T be the output vector
of the real system.
Mathematical Problems in Engineering
7
In the traditional trajectory sensitivity approach, the following auxiliary system would
be chosen to estimate parameters of Lorenz’s system:
ż1 −σz1 σz2 ,
ż2 −z2 − z1 z3 rz1 ,
3.9
ż3 −bz3 z1 z2 .
This system has exactly the same structure as Lorenz’s system model; the only
difference is that the state variables xi , i 1, 2, 3 were replaced by zi , i 1, 2, 3, and
the parameters do not have the subindex r used for the real parameters. Assuming that
measurements are subject to random noise, the initial conditions of states zi , i 1, 2, 3
are unknown and have to be estimated. Thus, the extended parameter vector is pr σr , rr , br , z1o , z2o , z3o .
Our experience shows that Lorenz’s system parameters cannot be simultaneously
estimated using the traditional trajectory sensitivity methodology even for very small
displacements of the initial conditions and parameter values from real values. Nonlinear
systems with chaotic behavior, such as Lorenz’s system, have a very small convergence
region, because of their high relative output sensitivities with respect to the parameters.
Figure 3 shows the sensitivities of output 1 with respect to Lorenz’s system
parameters. The sensitivity of output z1 with respect to parameter σ is very small when
compared with sensitivities of the z1 output with respect to parameter b or r. Such behavior
makes the approach of Figure 1 inappropriate for parameter estimation of chaotic systems.
In the particular Lorenz’s system case, errors as small as ±1% in the initial parameter and
initial condition guesses lead the algorithm to nonconvergence due to lack of synchronization
between the real system and the auxiliary system.
A master-slave coupling between the real system and the auxiliary system will be used
to enhance the numerical robustness of the trajectory sensitivity-based parameter estimation
algorithm. For this purpose, the next auxiliary system will be employed:
ż1 −σz1 σz2 ,
3.10
ż2 −z2 − x1 z3 rx1 ,
3.11
ż3 −bz3 x1 z2 ,
3.12
where z z1 , z2 , z3 T is the state space vector. In this case, the states will be taken as
the outputs of the auxiliary system, that is, y z1 , z2 , z3 T . The auxiliary system 3.10 to
3.12 resembles Lorenz’s system; the difference is that state variable z1 was replaced by the
coupling variable x1 in some convenient positions, in order to decrease nonlinearities of the
auxiliary system.
The replacing of the coupling variable is accomplished in order to eliminate the
nonlinear terms in the difference system 3.13. The substitution of the coupling variable x1 in
3.10 does not have any influence in the estimate process, for being a linear term. However,
we still do not have a systematic procedure to determine the best terms to substitute.
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Mathematical Problems in Engineering
Sensitivities
400
200
0
−200
−400
−600
−800
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time s
∂z1 /∂σ
∂z1 /∂r
∂z1 /∂b
Figure 3: Z1 output sensitivity Lorenz’s system with respect to the parameters σ, r, and b.
Real system
x1
x1
x1
x2
Auxiliary system
ż1 σz2 − σz1
ż2 rx1 − z2 − x1 z3
ż3 x1 z2 − bz3
Sensitivity
equations
−
z1
z2
Parameter fitting
algorithm
pk1 pk − Γ−1 p
∂Jp
∂p
Figure 4: Master-slave trajectory sensitivity-based framework for parameter estimation of Lorenz’s system
parameters.
Although there is no rules in how to do this replacement, our experience shows
that these modifications enlarge the stability region of the parameter estimation algorithm.
Figure 4 illustrates the schematic diagram with this approach.
The error ei zi − xi , i 1, 2, 3 is derived from the difference between the outputs of
both the auxiliary and the real systems:
ė1 −σe1 σe2 x2 − x1 σe ,
ė2 −e2 − x1 e3 x1 re ,
3.13
ė3 x1 e2 − be3 − x3 be ,
where σe σ − σr , re r − rr , and be b − br are the error of parameter estimation.
It is very difficult to prove assumption A2. However, it will be shown that
assumption A1 is satisfied, for this choice of auxiliary system, in a different sense of
synchronization.
Mathematical Problems in Engineering
9
Table 1: Parameter estimation of Lorenz’s system using trajectory sensitivity method including masterslave coupling with noise in the measurements.
Parameter
σ
r
b
z1o
z2o
z3o
Deviation %
−54%
−54%
54%
5%
5%
−54%
Initial value
4.60
9.20
4.10
1.90
2.85
3.22
Estimated value
10.034
19.96
2.66
2.07
2.94
6.96
True value
10.0
20.0
2.66
2.00
3.00
7.00
Error %
0.34
−0.2
0.00
3.5
2.0
−0.57
Definition 3.1. The outputs of the real system wt and the auxiliary system yt C1 synchronize in the interval Ta , Tb with precision ε if
sup wt − yt sup ẇt − ẏt < ε.
Ta ≤t≤Tb
Ta ≤t≤Tb
3.14
It is possible to prove, using 3.13 and The Implicit Function Theorem, that C1 synchronization, in the sense of Definition 3.1, implies that the estimated parameters are εclose to the real parameters.
Although assumption A2 cannot be easily proven, it can be easily checked at the
end of iterations. Moreover, the proposed master-slave coupling brings more robustness to
synchronization. Lorenz’s system parameters were estimated with good accuracy even for
cases where the first guess of parameter values was displaced up to ±54% from the real value.
Table 1 shows the estimation results under the presence of a ±5% white Gaussian random
noise in all the measurements.
The outputs of the real and the auxiliary system at the beginning and at the end of
iterations are shown in Figure 5.
3.2. Chua’s System
Chua’s circuit is a singular example of chaotic system, because it is the simplest circuit that
exhibits this kind of phenomenon. It is composed only by one inductor, two capacitors, one
resistor, and one nonlinear active resistor with a three-segment picewise-linear volt-current
V-I characteristics, called “Chua‘s Diode,” as shown in Figure 6.
Let the dimensionless equations of Chua’s circuit 18 represent the real system:
ẋ1 αr x2 − x1 − fx1 ,
3.15
ẋ2 x1 − x2 x3 ,
3.16
ẋ3 −βr x2 − γr x3 ,
3.17
1
fx1 bx1 a − b{|x1 1| − |x1 − 1|},
2
3.18
where α, β, γ, a, and b are the parameters to be estimated. We assume that states are
measured; that is, w x1 , x2 , x3 T is the output vector of the real system. Assuming that the
10
Mathematical Problems in Engineering
Output 3 at the beginning of iterations
35
30
Output 3 at the end of iterations
30
25
Amplitude
Amplitude
25
20
15
20
15
10
10
5
5
0
0
0.5
1
1.5
2
0
0
0.5
Time s
1
1.5
2
Time s
x3 , real system
z3 , auxiliary system
x3 , real system
z3 , auxiliary system
a
b
Figure 5: Comparison between the outputs of the auxiliary system at the beginning and at the end of
iterations for output number 3. A white Gaussian noise with zero mean and standard deviation of 5% of
the highest measured value was applied to all measurements.
initial conditions of differential equations of the model are unknown and they also have to be
estimated. Thus, the extended parameter vector is p α, β, γ, a, b, x1o , x2o , x3o , whose true
values are αr 6.5792, βr 10.9024, γr −0.0445, ar −1.1829, br −0.6524, x1o 0.15,
x2o 0.90, and x3o 0.80. Measurements of the real system were obtained by numerical
integration of 3.15 to 3.18, with their true parameters and initial conditions. Random white
Gaussian noise of ±5% of the peak value was added to the real measurements.
Like Lorenz’s system, parameters of Chua’s circuit cannot be simultaneously
estimated using the traditional trajectory sensitivity approach of Figure 1 due to high relative
sensitivity of trajectories with respect to parameters and initial conditions. Our experience
shows that errors as small as ±1% on the initial parameter guesses lead the algorithm to
nonconvergence due to lack of synchronization.
In order to overcome this difficulty, the following auxiliary system was chosen:
ż1 α z2 − x1 − fx1 − kz1 − x1 ,
ż2 x1 − z2 z3 ,
ż3 −βz2 − γz3 ,
3.19
1
fx1 bx1 a − b{|x1 1| − |x1 − 1|}.
2
The auxiliary system resembles Chua’s circuit model; the difference is that a masterslave coupling between the real and the auxiliary systems was employed; that is, state
variable z1 was replaced by the measured real variable x1 in some convenient positions
to reduce nonlinearities in the auxiliary system. Moreover, to enhance synchronization
robustness, an extra term was added to 3.15, gz1 , x1 −kz1 − x1 , where k 10. Figure 7
illustrates this parameter estimation scheme.
Mathematical Problems in Engineering
11
R
lnr
Gb
L
C1
C2
−B
NR
Ga
Vnr
B
Gb
r0
a
b
Figure 6: Chua’s circuit and the three-segment odd-symmetric picewise-linear volt-current V-I characteristic of Chua’s diode.
Real system
x1
x1
x2
x3
−
Auxiliary system
ż1 αz2 − x1 − fx1 − kz1 − x1 ż2 x1 − z2 z3
ż3 −βz2 − γz3
1
fx1 bx1 a b|x1 1| − |x1 − 1|
2
x1
Sensitivity
equations
z1
z2
z3
Parameter fitting
algorithm
pk1 pk − Γ−1 p
∂Jp
∂p
Figure 7: Master-slave trajectory sensitivity based framework for parameter estimation of Chua’s circuit.
The error ei zi − xi , i 1, 2, 3 is derived from the difference between the outputs of
both the auxiliary and the real systems:
ė1 −be1 αe2 αe x2 − x1 − x1 αbe − br αe 1
− αe ar − br − αae − be {|x1 1| − |x1 − 1|},
2
3.20
ė2 −e2 e3 ,
ė3 −βe2 − γe3 − x3 γe − x2 βe ,
where αe α − αr , βe β − βr , γe γ − γr , ae a − ar , and be b − br are the error of
parameter estimation.
It is practicable proved that C1 -synchronization of outputs, in the sense of
Definition 3.1, implies that parameters of the auxiliary system p α, β, γ, a, b are ε-close
12
Mathematical Problems in Engineering
Outputs x2 real and z2 auxiliary
Outputs in the first iteration
2
1
0
−1
−2
0
0.02
0.04
0.06
0.08
0.1
Time s
Real system
Auxiliary system
a
Outputs x2 real and z2 auxiliary
Outputs at the end of iterations
1
0.5
0
−0.5
−1
0
0.02
0.04
0.06
0.08
0.1
Time s
Real system
Auxiliary system
b
Figure 8: Comparison between outputs of real and auxiliary systems at the beginning and at the end of
iterations for the output number 2. A white Gaussian noise with mean zero and standard deviation of 5%
of the highest measurement value was applied to all measurement.
to the real parameter values pr αr , βr , γr , ar , br ; that is, assumption A1 is satisfied. For
this purpose,the Implicit Function Theorem and 3.20 must be used.
Although assumption A2 cannot be easily proven, it can be checked at the end of the
iterations. Chua’s circuit parameters were successfully estimated even for parameter initial
guess deviations as large as 65% from the real values. Table 2 shows the estimation results.
The output of the real and the auxiliary system at the beginning and at the end of
iterations are shown in Figure 8.
Mathematical Problems in Engineering
13
Table 2: Parameter estimation of Chua’s system using trajectory sensitivity method including master-slave
coupling with noise in the measurements.
Parameter
α
β
γ
a
b
x0
y0
z0
Initial value
2.3027
3.8158
−0.0156
−0.4137
−0.2283
0.0525
0.3150
0.2800
Deviation
−65%
−65%
−65%
−65%
−65%
−65%
−65%
−65%
Estimated value
6.5435
10.8906
−0.0444
−1.1807
−0.6539
0.1542
0.9009
0.8185
True value
6.5792
10.9024
−0.0445
−1.1820
−0.6524
0.1500
0.9000
0.8000
Error %
0.54
0.11
0.27
0.12
0.24
2.83
0.10
2.31
4. Conclusions
A combination of trajectory sensitivity method and master-slave synchronization was
proposed to parameter estimation of nonlinear systems. It was shown that master-slave
coupling increases the robustness of the trajectory sensitivity algorithm with respect to the
initial guess of parameters. Since synchronization is not a guarantee that the estimation
process converges to the correct parameters, a conditional test that guarantees the new
combined methodology estimates that the true values of parameters was proposed. This
conditional test was successfully applied to Lorenz’s and Chua’s systems and the proposed
parameter estimation algorithm has shown to be very robust with respect to parameter initial
guesses and measurement noise for these examples.
Acknowlegments
This research was supported in part by Fundação de Amparo a Pesquisa do Estado de
São Paulo FAPESP and Conselho Nacional de Desenvolvimento Cientfico e Tecnológico
CNPq.
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